Aging properties of the voter model with long-range interactions

We investigate the aging properties of the one-dimensional voter model with long-range interactions in its ordering kinetics. In this system, an agent, S i = ± 1 , positioned at a lattice vertex i , copies the state of another one located at a distance r , selected randomly with a probability P ( r ) ∝ r − α . Employing both analytical and numerical methods, we compute the two-time correlation function G ( r ; t , s ) ( t ⩾ s ) between the state of a variable S i at time s and that of another one, at distance r , at time t . At time t , the memory of an agent of its former state at time s , expressed by the autocorrelation function A ( t , s ) = G ( r = 0 ; t , s ) , decays algebraically for α > 1 as [ L ( t ) / L ( s ) ] − λ , where L is a time-increasing coherence length and λ is the Fisher–Huse exponent. We find λ = 1 for α > 2, and λ = 1 / ( α − 1 ) for 1 < α ⩽ 2 . For α ⩽ 1 , instead, there is an exponential decay, as in the mean field. Then, in contrast with what is known for the related Ising model, here we find that λ increases upon decreasing α . The space-dependent correlation G ( r ; t , s ) obeys a scaling symmetry G ( r ; t , s ) = g [ r / L ( s ) ; L ( t ) / L ( s ) ] for α > 2. Similarly, for 1 < α ⩽ 2 , one has G ( r ; t , s ) = g [ r / L ( t ) ; L ( t ) / L ( s ) ] , where the length L regulating two-time correlations now differs from the coherence length as L ∝ L δ , with δ = 1 + 2 ( 2 − α ) .